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KERALA SCHOOL OF ASTRONOMY
AND
DEVELOPMENT OF CALCULUS
M.D.SRINIVAS
CENTRE FOR POLICY STUDIES
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BACKGROUND TO THE DEVELOPMENT
OF CALCULUS (c.500-1350)
Zero, infinitesimals and infinity
Background1)The concept ofpra - ntimantra ofopaniad.2)The concept of lopa in Pini, abhva in Nyya and nya in
Bauddha philosophy.3)nya as symbol in Chanda Stra (VIII.29) of Pigala (c.300BC)
Brahmagupta (c.628) on the mathematics of zero. The notion oftaccheda.
Bhskara II (c.1150) on the notion ofkhahara or ananta
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Zero, infinitesimals and infinity
Bhskarcrya, while discussing the mathematics of zero inLlvat, notes
that when further operations are contemplated, the quantity beingmultiplied by zero should not be changed to zero, but kept as is. Further,when the quantity which is multiplied by zero is also divided by zero, then
it remains unchanged. He follows this up with an example and declares
that this kind of calculation has great relevance in astronomy.
What is the number which when multiplied by zero, being added to half ofitself multiplied by three and divided by zero, amounts to sixty-three?
[ - ]
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Zero, infinitesimals and infinity
Bhskara, it seems, had not fully mastered this kind of "calculation withinfinitesimals" as is clear from some of the examples he considers in
Bjagaita, while solving quadratic equations by eliminating the middle
term (ekavara-madhyamharaa).
[ ]
[{0 (x + (x/2)}2
+ 2 {0(x + (x/2))}]/ 0 = 15.Bhskara in his Vsan just cancels out the zeroes and obtainsx = 2.
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Irrationals and iterative approximations
Background1)ulva-stra approximation for square-root of 2.2)ulva-stra approximation for .
Square-root algorithm in ryabhaya (Gaita 4) of ryabhaa(c.499). rdhara (c.850) on approximation of square-roots (Triatik
46).
Nryaa Paita (c.1356) on approximating square-roots by thesolutions ofvarga-prakti (Bjagaitvatasa 88).
Approximate value of (accurate to four decimal places) inryabhaya (Gaita 11). Successive approximations perhaps by
doubling of the circumscribing square, octagon etc. doubling, andcutting of corners (explained in Yuktibhand Kriykramakar).
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Second-order differences and interpolation in computation of
Rsines
Computation of Rsine-table (accurate to minutes in a circle ofcircumference 21,600 minutes) by the method of second-order
Rsine-differences in ryabhaya (Gtik 12, Gaita 12) of
ryabhaa (c.499).
Bj = Rsin (jh)
j = Bj+1 - Bj
j+1 - j = - Bj [(1 - 2)/ B1]
-Bj / B1
Second-order interpolation formula for finding arbitrary Rsinevalues in Khaakhdyaka (c. 665) of Brahmagupta.
Rsin (jh+
) =Bj + (/h) [(1/2)(
j +
j+1)(
/h)(
j-
j+1)/2]=Bj + (/h)(j+1 + j)/2 + (/h)2 (j+1 - j)/2
=Bj + (/h) j+1 + (/h) ((/h)-1)(j+1 - j)/2
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Summation of infinite geometric series
The geometric series 1 + 2 + ... 2n is summed in PigalasChanda-stra (c.300 BCE). Pigala also gives an algorithm forevaluating a positive integral power of a number in terms of anoptimal number of squaring and multiplication operations.
Mahvrcrya (c.850), in his Gaita-sra-sagraha gives thesum of a geometric series.
Vrasena (c. 816), in his Commentary Dhaval on theakhagama, has made use of the sum of the followinginfinite geometric series in his evaluation of the volume of thefrustum of a right circular cone:
1/4 + (1/4)2 + ... (1/4) n + .... = 1/3
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Ttklika-gati: Instantaneous velocity of a planet
Approximate formula for velocity (manda-gati) in terms ofRsine-differences was given by Bhskara I (c.630) and he alsocomments on its limitation (Laghu-bhskarya 2.14-15).
True velocity (sphua-manda-gati) in terms of Rcosine (as thederivative of Rsine) is given Laghu-mnasa (2.7) of Mujla (c.
932) andMah-siddhnta (3.15) ofryabhaa II (c. 950).
Bhskara II (c.1150) discusses the notion of instantaneousvelocity (ttklika-gati) and contrasts it with the so-called true
daily motion. He also evaluates the manda-gati and
ghra-gati(Vsan on Siddhnta-iromai 2.37-39).
Bhskara II notes the relation between maximum equation ofcentre (correction to displacement) and the vanishing of velocity
correction (Vsan on Siddhnta-iromai, Gola 4.3).
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Ttklika-gati: Instantaneous velocity of a planet
[ .- ]
[ . ]
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Surface area and volume of a sphere
In ryabhaya (Gaitapda 7), the volume of a sphere wasincorrectly estimated as the product of the area of a great circleby its square-root.
Bhskarcrya II (c.1150) has given the correct relation betweenthe diameter, the surface area and the volume of a sphere in his
Llvat.
In his V
san
commentary on Siddh
nta-
iromai
Bhskara hasalso presented justifications for these results. The surface area is
obtained by adding the areas of the sectors into which thesurface of the sphere is divided by a number of great circlesdrawn at equal distance. Volume of the sphere evaluated by
summing the volumes of pyramids with apex at the centre.
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NRYAA PAITA ON VRASAKALITA (c.1350)
ryabhaya, gives the sum of the sequence of natural numbers
1 + 2 + ... + n = n(n+1)/2
As also the result of first order repeated summation:
1.2/2 + 2.3/2 + ... + n(n+1)/2 = n(n+1)(n+2)/6
ryabhaas result for repeated summation was generalised to arbitraryorder by Nrayaa Paita (c.1350):
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THE KERALA SCHOOL OF ASTRONOMY (c.1350-1825)
Kerala traces its ancient mathematical traditions to Vararuci. There are
speculations that
ryabhaa hailed from Kerala. In the classical period,there were many great Astronomer-Mathematicians in Kerala such as
Haridatta (c.650-700), Devcrya (c.700), Govindasvmin (c.800),akaranryaa (c.850) and Udayadivkara (c.1100). However it was
Mdhava of Sagamagrma (near Ernakulam) who pioneered a new
School of Astronomy and Mathematics
Mdhava (c.1340-1425): Vevroha, Sphuacandrpti, and a few tracts
are all that is available apart from citations in later works.
Paramevara of Vaasseri (c.1360-1455), a disciple of Mdhava:
Dggaita, Goladpik, and commentaries on Sryasiddhnta,
ryabhaya, Mahbhskarya, Laghubhskarya, Laghumnasa and
Llvat and Siddhntadpik on Govindasvmins commentary onMhabhskarya.
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THE KERALA SCHOOL OF ASTRONOMY (c.1350-1825)
Nlakaha Somayjof Tkaiyr (c.1444-1555), student of Dmodara son
of Paramevara: Tantrasagraha, ryabhayabhya, Golasra,
Candracchygaita, Siddhntadarpaa, Jyotirmms and Grahasphu-
nayane Vikepavsan.
Jyehadeva of Parakroa (c.1500 - 1610), student of Dmodara: Gaita-
Yukti-Bh (c. 1530 in Malaylam),Dkkaraa.
Citrabhnu (c.1475-1550), student of Nlakaha: Karamta,
Ekaviatipranottara.
akara Vriyr of Tkuaveli (c.1500-1560), student of Citrabhnu:
Karaasra, commentaries Kriykramakar (c.1535) onLlvat, Yukti-dpik,
Kriykalpa (in Malaylam) andLaghuvivti on Tantrasagraha.Acyuta Pirai (c.1550-1621), student of Jyehadeva and teacher of
Nryana Bhatiri: Sphuanirayatantra, Karaottama, Rigola-sphuanti,
Malaylam commentary on Vevroha and a few tracts.
The school continued to flourish till early nineteenth century. Some of theimportant works are Karaapaddhati (c.1700?) of Putumana Somayj and
Sadratnamlofakaravarman (c.1774-1839).
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NLAKAHA ON THE IRRATIONALITY OF
One of the main motivations of the mathematical work of the Kerala
school is paridhi-vysa sambandha, obtaining accurately the relationbetween the circumference of a circle. ryabhaa (c.499) had given the
following approximate value for:
[ ]One hundred plus four multiplied by eight and added to sixty-two
thousand: This is the approximate measure of the circumference of a circle
whose diameter is twenty-thousand.
Thus, according to ryabhaa, 62832/20000=3.1416Nlakaha Somayaji in his ryabhaya-bhya, while discussing square-roots, explains that only the approximate value is given for as thetraditional methods for its evaluation of involve square-roots:
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NLAKAHA ON THE IRRATIONALITY OF
Later, Nlakaha states that the ratio of the circumference to the diameter
of a circle cannot be expressed as the ratio of two integers exactly.
?
...
"Why then has an approximate value been mentioned here instead of the
actual value? This is the explanation. Because the actual value cannot be
expressed. Why? Given a certain unit of measurement in which thediameter has no fractional part, the same measure when applied to
measure the circumference will certainly have a fractional part. ...Thus
when both are measured by the same unit they cannot both without
fractional parts. Even if you go a long way (by choosing smaller andsmaller units of measure) a small fractional part will remain. The import
[ofsanna] is that there will never be a situation where both are integral."
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NLAKAHA ON THE SUM OF INFINITE GEOMETRIC SERIES
Vrasena (c. 816), had made use of the sum of the following infinite
geometric series1/4+(1/4)
2+ (1/4)
n+. =1/3
This is proved in the ryabhaya-bhya by Nlakaha Somayj, who
makes use of this series for deriving an approximate expression for a small
arc in terms of the corresponding chord in a circle. Nlakaha begins hisdiscussion of the sum of the infinite geometric series by posing the issue
as follows:
?
"The entire series of powers of 1/4 adds up to just 1/3. How is it
known that [the sum of the series] increases only up to that [limiting
value] and that it actually does increase up to that [limiting value]?"
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NLAKAHA ON THE SUM OF INFINITE GEOMETRIC SERIES
Nlakaha obtains the sequence of results
1/3 = 1/4 + 1/(4.3)1/(4.3) = 1/(4.4) + 1/(4.4.3)1/(4.4.3) = 1/(4.4.4) + 1/(4.4.4.3)
and so on, from which he derives the general result
1/3 - [1/4 + (1/4)2 + ... + (1/4)n] = (1/4n)(1/3)Nlakaha then goes on to present the following crucial argument toderive the sum of the infinite geometric series: As we sum more terms, the
difference between 1/3 and sum of powers of 1/4 (as given by the right
hand side of the above equation), becomes extremely small, but neverzero. Only when we take all the terms of the infinite series together do we
obtain the equality
1/4 + (1/4)2 + ... + (1/4)n + ... = 1/3
Nlakaha uses the above series to prove the following relationbetween the cpa (arc), jy (Rasine and ara (Rversine) for small arc:
Cpa [(1+1/3) ara2 + Jy2 ]1/2
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BINOMIAL SERIES EXPANSION
In obtaining the accurate relation between circumference and diameter, the
binomial series expansion plays an important role.
Given three positive numbers a, b, c, with b > c. we have the identity
Now substituting for (b-c)/b on the right from the equation and iteratingwe get
If we set [(b-c)/c] =x, the above is the well-known binomial series
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BINOMIAL SERIES EXPANSION
The derivation of the binomial series as well as the other results that we
discuss are found in Gaita-Yukti-Bha and Kriykramakar.
As regards the binomial series they note that there is no logical end to the
process of iterations and that one may stop after having obtained results tothe desired accuracy when the later terms will only get smaller and
smaller. They also note that the latter will happen only when (b-c) < c
(which is the condition for the convergence of the binomial expansion).
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SUM OF INTEGRAL POWERS OF NATURAL NUMBERS
The derivation of the Mdhava series for also involves estimating, for
large n, the value of the sama-ghta-sakalita
Sn(k)=1k + 2k + ... n k
Explicit formulae were given by ryabhaa for k = 1, 2 and 3.
Sn(1)=1+2+...n
=n(n+1)/2
Sn(2)=1
2+2
2+... +n
2=n(n+1)(2n+1)/6
Sn(3)=1
3+2
3+... +n
3=[1
+2+... +n]
2 =[n(n+1)/2]
2
Gaita-Yukti-Bh and Kriykramakarderive the following estimate forthe general sama-ghta-sakalita:
Sn(k)= 1
k+2
k++n
k nk+1/(k+1) forlargen
They also give an estimate for the repeated summation (vra-sakalita)
Vn(1)=1+2+3+...+n =n(n+1)/2
Vn(k)=V1
(k1)+V2
(k1)++Vn
(k1) n
k+1/[(k+1)!] forlargen
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MDHAVA SERIES FOR
The following verses of Mdhava are cited in Yuktibh and
Kriykramar
, which also present a detailed derivation of the relationbetween diameter and the circumference:
The first verse gives the Mdhava (Leibniz) series
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MDHAVA SERIES FOR
Mdhava also gave the cp-karaa series (Gregory Series) giving the cpa
(arc) associated with any jy
(Rsine) [It is also noted that we must ensurethat numerator < denominator in each term]:
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MDHAVA SERIES FOR
By using the cpkaraa series for an arc equal to one-twelfth of the
circumference (30), Mdhava gets a different series (later discovered byAbraham Sharp) for the ratio of the circumference to the diameter:
For an arc s which is one-twelfth of the diameter
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END CORRECTION TERMS
The Mdhava series for the circumference of a circle (in terms of odd
numbersp = 1, 3, 5, ...)C=4d[11/3+....+ (1)
(p1)/21/p+...]
is an extremely slowly convergent series. In order to facilitate
computation, Mdhava has given a procedure of using end-correction
terms (antya-saskra), of the form
In fact, the famous verses of Mdhava, which give the relation between
the circumference and diameter, also include the end-correction term
C=4d[11/3+....+...+(1)(p1)/2
1/p
+ (1)(p+1)/2{(p+1)/2}/{(p+1)2+1}]
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END CORRECTION TERMS
Mdhava has also given a finer end-correction term
C=4d[11/3+....+...+(1)(p1)/2
1/p
+(1)(p+1)/2[{(p+1)/2}2+1]/[{(p+1)2+5}{(p+1)/2}]
To Mdhava is attributed a value of accurate to eleven decimal places
which is obtained by just computing fifty terms with the above correction.
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MDHAVA CONTINUED FRACTION FOR
Both Yuktibh and Kriykramargive a derivation of the end correction
terms given by Mdhava, which involve a careful estimate of the errorinvolved in terms of inverse powers of comparison of the odd numberp.
By carrying this process further, we find that the end-correction term 1/apcan be expressed as a continued fraction:
Using the above correction term forp = 1, we get what may be called the
Mdhava continued fraction for :
2/(4 )=2+12/2+ 2
2/2+ 3
2/2+...
Brouncker Continued Fraction (1656):
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TRANSFORMED SERIES FOR
Adding and subtracting the end-correction terms, we can write the
Mdhava series for in the form:
By choosing different correction terms we get different transformed series
many of which are also converge faster than the Mdhava (Leibniz) series.
If we choose what may be termed the zero-th order correction divisor,
ap = 2p+2, we get the series
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TRANSFORMED SERIES FOR
If we choose the first-order correction divisor given by Mdhava,
then we get the series
:
Yuktibh and Kriykramardo not discuss the transformed series when
we use the more accurate correction divisor given by Mdhava. We caneasily see that it involves the seventh powers of the odd numbers.
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A HISTORY OF APPROXIMATIONS TO
Approximation to Accuracy
(Decimalplaces)
Method Adopted
Rhind Papyrus - Egypt
(Prior to 2000 BCE)
256/81 = 3.1604 1 Geometrical
Babylon (2000 BCE) 25/8 = 3.125 1 Geometrical
ulva Stras (Prior to
800 BCE)
3.0883 1 Geometrical
Jaina Texts (500 BCE) (10) = 3.1623 1 Geometrical
Archimedes (250 BCE) 3 10/71 < < 3 1/7 2 Polygon doubling(6.2
4= 96 sides)
Ptolemy (150 CE) 3 17/120 = 3. 141666 3 Polygon doubling(6.2
6= 384 sides)
Lui Hui (263) 3.14159 5 Polygon doubling
(6.29
= 3072 sides)
Tsu Chhung-Chih
(480?)
355/113 = 3.1415929
3.1415927
6
7
Polygon doubling
(6.29
= 12288
sides)
ryabhaa (499) 62832/20000 = 3.1416 4 Polygon doubling(4.2
8= 1024 sides)
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A HISTORY OF APPROXIMATIONS TO
Approximation to Accuracy
(Decimalplaces)
Method Adopted
Mdhava (1375) 2827433388233/9.1011
= 3.141592653592
11 Infinite series with
end corrections
Al Kashi (1430) 3.1415926535897932 16 Polygon doubling (6.227
sides)
Francois Viete (1579) 3.1415926536 9 Polygon doubling (6. 216
sides)
Romanus (1593) 3.1415926535..... 15 Polygon doubling
Ludolph Van Ceulen
(1615)
3.1415926535..... 32 Polygon doubling (262
sides)
Wildebrod Snell
(1621)
3.1415926535..... 34 Modified polygon doubling
(2
30
sides)Grienberger (1630) 3.1415926535..... 39 Modified polygon doubling
Isaac Newton (1665) 3.1415926535..... 15 Infinite series
Abraham Sharp (1699) 3.1415926535..... 71 Infinite series for tan-1
(1/ 3)
John Machin (1706) 3.1415926535..... 100 Infinite series relation
/4 = 4 tan-1 (1/5)-
tan-1 (1/239)Ramanujan (1910,
1914) Gosper (1985)
17
Million
Modular Equation
Kondo, Yee (2010) 5 Trillion Modular Equation
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A HISTORY OF EXACT RESULTS FOR
Mdhava (1375) /4 = 1 1/3 + 1/5 1/7 + ...
/
12 = 1- 1/3.3 +1/3
2
.5 1/3
3
.7 +.../4 = 3/4 + 1/(33-3) 1/(53-5) + 1/(73-7) - .../16 = 1/(15+4.1) -1/(35+4.3) +1/(55+4.5) - ...
Francois Viete (1593) 2/ = [1/2] [1/2 + 1/2(1/2)] [1/2 + 1/2(1/2+1/2(1/2))]...(Infinite product)
John Wallis (1655) 4/ = (3/2)(3/4) (5/4)(5/6)(7/6)(7/8)... (Infinite product)
William Brouncker(1658)
4/ = 1+ 12/2+ 32/2+ 52/2+ ... (Continued fraction)
Isaac Newton (1665) = 3 3 /4 + 24 [1/3.8 1/5.32 1/7.128 1/9.512...]
James Gregory (1671) tan-1
(x) = 1x/3 +x2/5 - ...
Gottfried Leibniz
(1674)
/4 = 1 1/3 + 1/5 1/7 + ...
Abraham Sharp
(1699)
/ 12 = 1- 1/3.3 +1/32.5 1/33.7 +...
John Machin (1706) /4 = 4 tan-1 (1/5) - tan-1 (1/239)
Ramanujan (1910, 1914)
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RAMANUJANS SERIES FOR
One of Ramanujan's early papers is on the Modular equations and
approximations to
. Though published later from London in 1914 (QJM1914, 350-372), it is said to embody so much of Ramanujans early Indian
work. Here is a sample of his results:
Ramanujan also notes that the last series "is extremely rapidly convergent".
Indeed in late 1980s, it blazed a new trail in the saga of computation of.
MDHAVA SERIES FOR RSINE AND RVERSINE
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MDHAVA SERIES FOR RSINE AND RVERSINE
The jy or bhuj-jy of an arc of a circle is actually the half the chord
(ardha-jy
or jy
rdha) of double the arc. In the figure below, if r is theradius of the circle, jy (Rsine), koi or koi-jy (Rcosine) and ara
(Rversine) of the cpa (arc) EC = s= r, are given by:
jy(arcEC) =Rsin(s)=CD=r sin
koi(arcEC)=Rcos(s)=OD=rcos
ara(arcEC)=Rvers(s)=ED=rrcos
NLAKAHAS DERIVATION OF RYABHAA RELATION FOR
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NLAKAHAS DERIVATION OF RYABHAA RELATION FOR
SECOND ORDER SINE DIFFERENCES
We consider a given arc of arc-length s, which is divided into n equal arc-
bits. Ifs = r, then the j-th pia-jyBj and the corresponding koi-jy
Kj, and the ara Sj, are
Bj=Rsin(js/n)=rsin(j/n)=rsin(js/rn) [Cj Pj in the Figure]
Kj=Rcos(js/n)=rcos(j/n)=rcos(js/rn) [Cj Tj in the Figure]
Sj=Rvers(js/n)=r[1cos(j/n)]=r[1 cos(js/rn)] [Pj E in the Figure]
RYABHAA RELATION FOR SECOND ORDER SINE DIFFERENCES
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RYABHAA RELATION FOR SECOND-ORDER SINE-DIFFERENCES
Let M j+1 be the mid-point of the arc-bit CjCj+1 and similarly M j the mid-
point of the previous (j-th) arc-bit. We shall denote the pia-jy
of the arcEMj+1 as Bj+1/2 and clearly Bj+1/2 = Mj+1Qj+1. The corresponding
Kj+1/2 = Mj+1Uj+1 and Sj+1/2 = EQj+1. Similarly, Bj-1/2 = MjQj, Kj-1/2 = MjUj and
Sj-1/2 = EQj. The full-chord of the arc-bit s/n may be denoted . Then a
simple argument based on trair
ika (similar triangles) leads to therelations for Rsine and Rcosine differences
j=Bj+1Bj=( /R)Kj+1/2
Kj1/2Kj+1/2=(Sj+1/2Sj1/2)=( /R)Bj
Thus, we obtain the relation for second-order sine differences:
j+1 j = (Bj+1Bj)(BjBj1)= ( /R)2Bj=(2 1)Bj/B1
With n =24, ryabhaa used the approximation: (1 2) 1,B1225'
Nlakaha (Tantrasagraha): B1224'50",(1 2)/B11/233'30"
akara Vriyar (Laghuvivti):B1224'50"22"',(1 2)/B11/233'32"
MDHAVA SERIES FOR RSINE
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MDHAVA SERIES FOR RSINE
Yuktibh and Yuktidpik give the derivation of this and the Rversine
series by dividing the arc s into a large number n of equal parts and using
the relations between second-order Rsine differences (khaa-jyntara)
and the Rsines of arcs ns/j (piajys)-
MDHAVA SERIES FOR RVERSINE
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MDHAVA SERIES FOR RVERSINE
The verses giving the Rsine and Rversine series also note that the methodof obtaining accurate approximations to Rsine and Rversine values as
encoded in the mnemonics (also due to Mdhava) Vidvn etc and Stena
etc, indeed follow from these series.
NLAKAHAS FORMULA FOR INSTANTANEOUS VELOCITY
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NLAKAHA S FORMULA FOR INSTANTANEOUS VELOCITY
(c.1500)
Instead of basing the calculation of instantaneous velocity on theapproximate form of manda-correction, Nlakaha Somayj uses the
exact form of the manda correction
=m+Sin1[(r0/R)(1/R)Rsin(m)]
Nlakaha gives the correct formula for the correction to the meanvelocity in his treatise Tantrasagraha.
Nlakaha gives the derivative of the second term above in the form
[{(r0/R)Rcos(m)}/{R2(r0/R)
2Rsin
2(m)}
1/2][(d/dt)(m)]
ACYUTAS FORMULA FOR INSTANTANEOUS VELOCITY (c 1600)
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ACYUTA S FORMULA FOR INSTANTANEOUS VELOCITY (c.1600)
Acyuta Pirai in his Sphuaniraya-tantra gives the Nlakaha formula
for the instantaneous velocity. He also discusses an alternative prescription
for manda-correction due to Mujla (c.932) given by
=m+[(r/R)Rsin(m)]/[R(r/R)Rcos(m)]
Acyuta notes that in this model the manda-correction also depends on the
hypotenuse and hence the correction to the mean velocity is given by:
This gives the derivative of the second term above as
[{(r/R)Rcos(m)}{(r/R)Rsin(m)}2/{R(r/R)Rcos(m)}]
x [1/{R(r/R)Rcos(m)}] [(d/dt) (m)]
REFERENCES
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REFERENCES
1.Gaitayuktibh (c.1530) of Jyehadeva (in Malayalam):Gaitdhyya, Ed., with Notes in Malayalam, by Ramavarma
Thampuran and A. R. Akhileswara Aiyer, Trichur 1948. Ed. with Tr. byK.V.Sarma with Explanatory Notes by K Ramasubramanian, M DSrinivas and M.S.Sriram, 2 Volumes, Hindustan Book Agency, Delhi2008.
2.Kriykramakar (c.1535) of akara Vriyar on Llvat ofBhskarcrya II: Ed. by K.V. Sarma, Hoshiarpur 1975.
3.K.V.Sarma, A History of the Kerala School of Hindu Astronomy,Hoshiarpur (1972)
4.S. Parameswaran, The Golden Age of Indian Mathematics, SwadeshiScience Movement, Kochi 1998.
5.C.K.Raju, Cultural Foundations of Mathematics: The Nature ofMathematical Proof and the Transmission of the Calculus from India to
Europe in the 16thc.CE, Pearson Education, Delhi 2007.6.G.G.Joseph,A Passage to Infinity, Sage, Delhi (2009)7. K.Ramasubramanian and M.D.Srinivas, Development of Calculus in
India, in C.S.Seshadri (ed) , Studies in History of Indian Mathematics,Hindustan Book Agency, Delhi (2010), pp.201-286.
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